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Integrate[1/(x^3*Sqrt[(a^2 - x^2)^3]), x] ==
-((-a^2 + x^2)*(-a^3 + 3*a*x^2 + 3*x^2*Sqrt[a^2 - x^2]*Log[3*a*x] - 3*x^2*Sqrt[a^2 - x^2]* Log[4*a^5*(a + Sqrt[a^2 - x^2])]))/ (2*a^5*x^2*Sqrt[(a^2 - x^2)^3])

                   1
Integrate[-------------------, x] == 
           3        2    2 3
          x  Sqrt[(a  - x ) ]

     2    2     3        2
-((-a  + x ) (-a  + 3 a x  + 
 
          2       2    2
       3 x  Sqrt[a  - x ] Log[3 a x] - 
 
          2       2    2
       3 x  Sqrt[a  - x ] 
 
               5            2    2
        Log[4 a  (a + Sqrt[a  - x ])])) / 
 
      5  2        2    2 3
  (2 a  x  Sqrt[(a  - x ) ])